Abstract wrote:Berg and Ulfberg and Amano and Maruoka have used CNF-DNF-approximators to prove exponential lower bounds for the monotone network complexity of the clique function and of Andreev's function. We show that these approximators can be used to prove the same lower bound for their non-monotone network complexity. This implies P not equal NP.

Some years ago I went to this talk, where the idea was that an NMR spectrometer could implement a NAND gate, and so was in a sense a universal computer. The NMR machine was connected to a computer but we were supposed to believe that the computer, which was full of logic gates, wasn't doing any of the work. I had my doubts, and I think it highly likely that the computer was acting as a NOT gate.

My point is that it is all to easy to forget how powerful NOT is. Now there is a proof which says that a proof based on monotone boolean networks (which I understand to mean networks without NOT), can be twerked a bit to apply to networks which include NOT.